On the special values of Berndt's generalized $L$-function

In this paper, we study the generalized $L$-function considered by Berndt (1975). We introduce the generalized Apostol-Bernoulli polynomials and the generalized Apostol-Bernoulli functions, and establish some properties for them, including the Fourier series for these functions. We show that the values of Berndt’s generalized $L$-function at integers are explicitly evaluated in terms of the generalized Apostol-Bernoulli functions.